Pointwise Operation/Examples
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Examples of Pointwise Operations
Cube and Sine Functions
Let $f$ and $g$ be the real functions defined as
\(\ds \forall x \in \R: \, \) | \(\ds \map f x\) | \(=\) | \(\ds x^3\) | |||||||||||
\(\ds \forall x \in \R: \, \) | \(\ds \map g x\) | \(=\) | \(\ds \sin x\) |
Then for all $x \in \R$:
\(\ds \map {\paren {f + g} } x\) | \(=\) | \(\ds x^3 + \sin x\) | ||||||||||||
\(\ds \map {\paren {f - g} } x\) | \(=\) | \(\ds x^3 - \sin x\) | ||||||||||||
\(\ds \map {\paren {f \times g} } x = \map {\paren {g \times f} } x\) | \(=\) | \(\ds x^3 \sin x\) | ||||||||||||
\(\ds \map {\paren {f \times f} } x\) | \(=\) | \(\ds x^6\) |
Contrast this with:
\(\ds \map {\paren {f \circ g} } x\) | \(=\) | \(\ds \paren {\sin x}^3\) | ||||||||||||
\(\ds \map {\paren {g \circ f} } x\) | \(=\) | \(\ds \map \sin {x^3}\) | ||||||||||||
\(\ds \map {\paren {f \circ f} } x\) | \(=\) | \(\ds x^9\) |